TABLE OF PYTHAGORAS.

1

2

3

6 7

8

9 10 11 12

13 14 15 16

17

18

2

4

61

8 10

12

14

16

18 20 22 24

26 28 30 32 34

36

11 22 33 44 55

12 24 36 48 60
13 26 39 52 65
14 28 42 56 70

15 30 45 60 75

19 20 21 22 23 24 36 38 40 42 44 46 48 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 64 68 72 76 80 84 88 92 96 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 42 48 54 60 66 72 78 49 56 63 70 77

84

6 12 18 24 30 7 14 21 28 35 42 8 16 24 32 40 48 88 96 104 112 120 128 136|144 152 160 168 176 184 192 9 18 27 36 45 51 63 81 90 99 108 117 126 135 144|153|162 171|180 189 198 207 216 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 77 88 99 110 121 132 143 154 165 176 187 198 209 220 231 242 253 264 72 841 96 108 120 132|144|156 168 180 192 204 216 228 240 252|264|276|288 78 91|104|117|130 143 156 169 182 195 208|221 234 247 260 273 286|299|312 84 98112|126|140 154 168 182 196|210|224 238 252 266 280 294 308 322 336 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360 96 112 128 144 160 176 192 208|224 240 256 272 288|304|320 336|352 368 384 85 102|119 136 153 170 187|204 221 238 255 272 289 306|323 340 357 374 391 408 18 36 54 72 90 108|126|144|162|180|198 216 234|252 270 288 306 324 342|360|378|396|414|432 19 38 57 76 95|114|133|152 171|190 2091228|247|266|285|304|323|342|361 380 399 418 437 456| 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 21 421 63 84|105|126|147|168|189|210|231 252 273|294|315 336 357 378 399 420 441 462 483 504 22 44 66 S8|110|132 154 176|198|220|242 264 286|308) 330 352 374 396 418|440|462|484|506|528 23 46 69 92 115 138 161|184|207|230|253|276|299|322|345 368|391|414|437|460|483 506 529|552 24 48 72 96|120 144|168|192|216|240|264 288 312 336 360 384 408 432 456 480 504 528 552 576

90 96 102 108|114|120|126|132|138|144 84 91 98105|112 119 126 133 140 147 154 161 168

EXAMPLES.

3.

123456789

4.

987654321

6

6388888888

36. Multiply 7008005 by 10008. 37. Multiply 4001100 by 40506. 38. Multiply 6716700 by 808070. 39. Multiply 987648 by 481007. 40. Multiply 18711000 by 470. 41. Multiply 10000 by 7000.

Ans. 70136114040. Ans. 162068556600. Ans. 5427563769000. Ans. 475065601536. Ans. 8794170000. Ans. 70000000.

42. Multiply 101010101 by 2020202. Ans. 204060808060402.

43. Multiply 70000 by 10000.
44. Multiply 800008 by 9009.
45. Multiply 900900 by 70070.
46. Multiply 4807658 by 706007.
47. Multiply 16789001 by 10080.

Ans. 700000000. Ans. 7207272072. Ans. 63126063000. Ans. 3394240201606.

Ans. 169233130080.

48. Multiply 304050607 by 3011101. Ans. 915527086788307. 49. Multiply 908007004 by 500123. Ans. 454115186861492. 50. Multiply 2003007001 by 6007023.

Ans. 12032109124168023. 51. Multiply 9000006 by 9000006. Ans. 81000108000036. 52. Multiply 1152921504606846976 by 1152921504606846976. Ans. 1329227995784915872903807060280344576. 53. What will 27 oxen cost at 35 dollars each ?

Ans. $945. 54. What will 365 acres of land cost at 73 dollars per acre ? Ans. $26645.

55. What will 97 tons of iron cost at 57 dollars a ton?

Ans. $5529. 56. What will 397 yards of cloth cost at 7 dollars per yard? Ans. $2779.

57. What will 569 hogsheads of molasses cost at 37 dollars per hogshead? Ans. $21053. 58. If a man travel 37 miles in one day, how far will he travel in 365 days? Ans. 13505 miles. 59. If one quire of paper have 24 sheets, how many sheets are in a ream, which consists of 20 quires? Ans. 480 sheets. 60. If a vessel sails 169 miles in one day, how far will she sail in 144 days? Ans. 24336 miles.

61. What will 698 barrels of flour cost at 7 dollars a barrel? Ans. $4886.

62. What will 376 lbs. of sugar cost at 13 cents a pound?

Ans. 4888 cts.

63. What will 97 lbs. of tea cost at 93 cents a pound?

Ans. 9021 cts.

64? If a regiment of soldiers consists of 1128 men, how

many men are there in an army of 53 regiments?

Ans. 59784.

65. What will an ox weighing 569 pounds amount to at 8 cents a pound? Ans. 4552 cts. 66. If a barrel of cider can be bought for 93 cents, what will 75 barrels cost? Ans. 6975 cts. 67. If in a certain factory 786 yards of cloth are made in one day, how many will be made in 313 days? Ans. 246018 yds. 68. A certain house contains 87 windows, and each window has 32 squares of glass; how many squares are there in the Ans. 2784 squares. 69. There are 407 wagons each loaded with 30009 pounds of coal; how many pounds are there in the whole ? Ans. 12213663 pounds. 70. Multiply three hundred and seventy-five millions two hundred and ninety-six thousand three hundred and twenty-one, by seventy-nine thousand and twenty-four.

whole house?

Ans. 29657416470704.

71. What would be the cost of 687 fothers of lead at 73 dollars a fother? Ans. $50151.

SECTION V.

DIVISION.

THE object of Division is to find how many times one num

ber is contained in another.

Division consists of three principal parts; the Dividend, or number to be divided; the Divisor, or number by which we divide; and the Quotient, which shows how many times the dividend contains the divisor.

When the dividend contains the divisor an exact number of times, the quotient is expressed by a whole number. But when this is not the case, there will be a remainder, when the division has reached its limit, and this remainder placed above the divisor, with a horizontal line between them, will form a fraction, and should be written at the right hand of the quotient, and will be a part of it. See Example 2d, and note.

1. The Remainder may be considered a fourth term in Division, and it will always be of the same denomination with the dividend.

For the sake of convenience, Division has been divided into two kinds, Long and Short.

2. All questions in which the divisor is not more than 12 may be conveniently performed by Short Division; all others are better performed by Long Division.

SHORT DIVISION.

EXAMPLE.

1. Divide 948 dollars equally among 4 men.

Dividend.

Divisor 4)948

Quotient 237

In performing this question, inquire how many times 4, the divisor, is contained in 9, which is 2 times, and 1 remaining; write the 2 under the 9 and suppose 1, the remainder, to be placed before the next figure of the dividend, 4, and the number will be 14. Then inquire how many times 4, the divisor, is contained in 14. 2 remaining. Write the 3 under mainder, 2, to be placed before the 8, and the number will be 28. times 28 will contain the divisor. which we place under the 8. Thus we find each man receives 237 dollars.

It is found to be 3 times and the 4, and suppose the renext figure of the dividend, Inquire again how many It is found to be 7 times,

From the above illustration, we deduce the following

RULE.

Write down the dividend and place the divisor on the left, with a curved or perpendicular line drawn between them. Draw also a horizontal line under the dividend, then observe how many times the divisor is contained in the first figure or figures of the dividend (beginning at the left hand), and place the quotient figure directly under the right-hand figure of the part of the dividend that was taken. If there be no remainder, proceed to inquire how many times the divisor is contained in the next figure* of the dividend, and set down the result at the right hand of the quotient figure already obtained, or directly under the figure of the dividend, and continue the work in this manner until the whole dividend is divided. But if there be a remainder either in the first or any subsequent division, imagine the number denoting it to be placed directly before the next figure of the dividend, and ascertain the number of times the divisor is contained in the number thus formed, and place the

*If this figure be smaller than the divisor, it cannot contain it, and the figure to be placed in the quotient will be a cipher. Sometimes, as when we divide by 11 or 12, we may have two successive ciphers in the quotient, as when the divisor is 12 and the next two figures are 1's or 1 and 0. We are then obliged to proceed to a third figure in the dividend, before we can effect a proper division.